MMOW Chapter 1 - COMAP · prestigious award for a young mathematician, the Adams Prize, for the ... Maxwell and geometrician Sir William Hodge. Although cryptographers will use

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C HA P T E R

Secret Codesand the Powerof AlgebraLESSON ONE

Keeping Secrets

LESSON TWO

UGETGV EQFGU

LESSON THREE

Decoding

LESSON FOUR

Cracking Codes

LESSON FIVE

Illusive Codes

LESSON SIX

Matrix Methods

Chapter 1 Review

11Examination Copy © COMAP Inc. Not for Resale

Examination Copy © COMAP Inc. Not for Resale

41

SECRET CODES AND THE POWER OF ALGEBRA

Getting and sending information is a part of

everyone’s life. Sometimes the information is private.

If private information falls into the hands of the

wrong person, the results can be bad.

Personal information is not all that needs to be protected. This is the

Information Age and most information is stored and sent

electronically. Wealth is transferred from one bank to another over

phone lines. Private messages are sent by email from one computer

to another. Important documents, stored on disk, can be retrieved

anywhere in the world.

Secret codes are vital in a world that relies on electronic

communication. Companies hire people called cryptographers to

design secret codes to protect information.

In this chapter, you focus on mathematical models for coding. To

create a coding model you must do two things: describe a way to

code and describe a way to decode.



PREPARATION READING

Codes Are Everywhere

Welcome to the world of secret codes! MTUJ DTZJSOTD YMJ HMFQQJSLJ! Codes are all around you.Codes help make sure that you are charged the right

price when you buy an item in a store. They also help get yourmail to the right address quickly. Many codes help speed theprocessing of information.

Some codes are used to keep information secret. People thinkof secret codes as tools of secret agents, but other people usethem. Banks, for example, keep their coding systems secret sothat thieves cannot access accounts. Secret codes are usedwhen a person wants to send a message and does not want toworry about others reading it.

How do you code a message so that it is easy to decode andhard to crack? People use mathematics to make secret codes. Inthis chapter, you will use mathematics as a tool to design goodmodels for coding.

You can think of the world of secret codes as having three keyroles: the coder, the decoder, and the code cracker. At times inthis chapter, you will play each role.

The message below has information that can help make andbreak secret codes. A mathematical process was used to codethe message. By the time you finish this chapter, you will beable to decode it.

I Y P K P I I P C F P I Q C B T D F D C P I Y P N T F I J T N NT Q K P I I P C F B Q I Y P P Q V K B F Y K D Q V L D V P

Once again, MTUJ DTZ JSOTD YMJ HMFQQJSLJ!

DISCUSSION/REFLECTION

1. Name some codes that you know. Is the code used to speedthe processing of information or to keep it secret?

2. Information is often sent in coded form so that a thirdparty cannot read it. What makes one coded messageharder to crack than another?

3. Who has the hardest job, the coder, the decoder, or the codecracker?

42 Preparation Reading

LESSON ONE

KeepingSecrets

Key Concepts

Coding

Decoding

Code breaking

Cryptography

Coding methods:substitutiontransportation

shift



43

The purpose of this activity is to help you decide what makes one codingprocess better than another.

In this activity, you play the role of code cracker and try to decode fivemessages. Each message is coded with a different process. You may notbe able to crack all of them now. But you will become a better codecracker as you go through the chapter.

For now, try to find the meaning of as many of the messages asyou can. Later you will have chances to revisit these messages and apply new techniques for cracking codes.

Message 1

BPWUIA RMNNMZAWV QVDMVBML I EPMMTTQSM LMDQKM BW KWLM UMAAIOMA

Message 2

6 19 26 18 7 10 23 8 6 17 17 10 9 6 8 13 10 8 16 9 14 12 14 25

14 24 26 24 10 9 25 20 9 10 25 10 8 25 10 23 23 20 23 24

14 19 8 20 9 10 24 17 14 16 10 31 14 21 8 20 9 10 24

Message 3

79 63 55 23 63 27 83 35 23 23 99 35 39 11 39 83 79 7 83 83 35 23

59 7 83 39 63 59 7 51 15 75 103 67 83 63 51 63 31 39 15

55 87 79 23 87 55 15 7 59 11 23 79 23 23 59 63 59 83 35 23

95 63 75 51 19 95 39 19 23 95 23 11

Message 4

HRKND JDNK GR UIBV EBHF HRLUBHG

KZDFD BAN BHGJBIIV BEIN GR HRAANHG

NAARAD

Activity 1.1 Lesson One

Activity 1.1: The World of Codes

In thiscourse, theproblems

you solve are as real as ispossible. However, youwould not be happy if themessages you had to cracktook months of effort. So, themessages in this and otheractivities are short. They arenot real historical messages,but they contain facts aboutthe use of codes in the world.

TAKE NOTE



44 Chapter 1 Secret Codes and the Power of Algebra Activity 1.1

Message 5

37 2 29 29 27 23 24 14 14 38 14 21 24 15 28 39 21 23

24 12 18 34 12 27 37 21 17 25 14 36 38 24 23 32 6 32

30 22 10 27 10 34 41 16 29 25 8 38 28 5 27 21 9 27

38 4 24 33 18 39 37 10 12 21 25 27 38 15 28 29 19 41

38 19 21 24 28 19 41 10 18

1. Try to crack one or more of the messages. Prepare topresent your results to the class.

2. Of the messages you cracked, which were the hardest?Give reasons for your answer.

3. What makes one coding model better than another?

Before you continue with your coding, decoding, and codecracking, you should know a few terms:

The original form of a message before it iscoded is called the plaintext.

Code and cipher can have different meaningsin some books, but they have the same mean-ing in this chapter. Both refer to the methodused to encode a message.

The process of putting a message in codedform is called encoding. Taking it from codedform back to plaintext is called decoding ordeciphering.

Finding the contents of a message withoutbeing told how it was coded is called codebreaking or code cracking.

Cryptography refers to the study of coding,decoding, and code breaking.

Secret codes have animportant role in history.Military commands werecoded during World Wars Iand II. The countries with thebest cryptographers held anadvantage in both wars. Thefamous Zimmermanntelegram was interceptedand decoded by the British.It revealed Germany’sattempt to ally with Mexicoagainst the United States inWorld War I. PresidentWoodrow Wilson asked for adeclaration of war shortlyafter this telegram wasintercepted.

As you can tell by looking atthe telegram, codedmessages are often long.Cracking them can takemonths of effort by a teamof cryptographers.

FYIExamination Copy © COMAP Inc. Not for Resale


Activity Summary

45Activity 1.1 Lesson One

In this activity, you:

� tried to decode five coded messages.

� explored what makes one coding process better than another.

By Karen Gold

Monday March 4, 2002

http://www.maths.nott.ac.uk/personal/sh/

A woman has won the UK’s mostprestigious award for a youngmathematician, the Adams Prize, for thefirst time in its 120-year history.

Dr. Susan Howson, 29, a Royal Societyfellow and lecturer at NottinghamUniversity, was lauded by the judges—an

international array of math professors—for her research on number theory andelliptic curves.

Previous winners of the £12,000 prize,awarded by Cambridge University,include the physicist James ClerkMaxwell and geometrician Sir WilliamHodge.

Although cryptographers will use Dr. Howson’s work, she is a puremathematician, choosing her subject“because of the beauty of the theorems.”

Lecturer Makes History in Math Examination Copy © COMAP Inc. Not for Resale


Individual Work 1.1: Sending Information

46

In this Individual Work you look at different types of codes. You alsoexplore three simple models for secret coding: the shift cipher, thesubstitution cipher, and the transposition cipher.

1. Recall that not all codes are used for secrecy. Some allow informationto be sent quickly and efficiently. For example, the U.S. Postal Serviceuses zip codes to speed the mail. (“Zip” stands for “zoneimprovement program.”) The simplest zip code is a five-digitnumber. (Some have nine digits; others have eleven.)

Zip codes are converted to a series of longand short bars that are read and sorted bya machine. The bars for several five-digitzip codes are shown in Figure 1.1.

a) Find the code for each digit from 0 to 9.

b) Why do you think the Postal Serviceuses five characters with three shortbars and two long ones? Could theyhave used two short bars and three long bars instead?

c) If the Postal Service used six bars with one long and five shorts ineach group, how many numerals could be coded?

d) With six bars, how many longs and shorts would have to be usedto cover ten numerals?

e) A good secret code is easy to encode, easy to decode, and hard to crack. Do you think that zip-code bars would make a goodsecret code?

2. Zip codes and universal product codes (UPCs) are common types ofcodes. Name other situations in which codes are used.

3. a) Write a short message. Design a secret code and use it to encodeyour message. Explain or diagram the coding process you used.

b) Challenge a member of your family or a friend to decipher yourcoded message. Describe the success or failure of the person whotried to crack your code. Note how long it took for the person tocrack the code or to give up.

4. Julius Caesar was the emperor of the Roman Empire during the firstcentury B.C. Caesar used a simple coding process called a shift cipherthat replaces each letter of the alphabet with a letter three placesremoved.

Chapter 1 Secret Codes and the Power of Algebra Individual Work 1.1

As you try to crack codedmessages, you might keepin mind the words ofCaptain Parker Hitt, whowrote the first U.S. Armymanual on code cracking:"Success in dealing withunknown ciphers ismeasured by these fourthings in the order named:perseverance, carefulmethods of analysis,intuition, and luck." Thischapter provides chancesto learn "careful methodsof analysis." You will have tosupply the perseverance.

FYI

Figure 1.1. The bars for four zip codes.



47Individual Work 1.1 Lesson One

A model for coding that uses addition or subtraction tomove or shift every letter of the alphabet to another letter in the alphabet is called a shift cipher. For example,a shift of +5 codes the letter C as the letter H because H isfive letters beyond C.

a) How does the shift (+5) cipher code the letters M and Y?

b) Use a shift (+8) cipher to code the first five words of this message:“Benedict Arnold used a substitution cipher to send messages tothe British in the Revolutionary War.”

5. For parts (a)–(d), use the given shift cipher to code the given letter.

a) shift (+5); Q b) shift (+11); W

c) shift (–8); N d) shift (+26); R

6. Another coding model is a substitution cipher. It replaces words orletters with other words or letters. The telegram in Figure 1.2 is anexample. It was sent to a Colonel Pelton about the 1876 presidentialrace. This message was coded by using replacements for certainsensitive words.

The coded message reads: Certificate required toMoses decision have London hour for Bolivia of justand Edinburgh at Moselle hand a any over GlasgowFrance received Russia of.

a) Make the following substitutions and write themessage that results: replace Bolivia withproposition, Russia with Tilden, London withcanvassing board, France with Governor Stearns,Moselle with two, Glasgow with hundred, Edinburghwith thousand, and Moses with Manton Marble.

b) The message in part (a) still doesn’t make sense because it wasalso coded with a transposition cipher. In a transpositioncipher, the order of the words or letters is changed.

The list below can help you unscramble the message. Forexample, the first word of the original message is theeighteenth word of the coded message, the second word ofthe original message is the twelfth word of the codedmessage, etc. (Hint: Governor Stearns, Manton Marble, andcanvassing board each count as single words.)

18, 12, 6, 25, 14, 1, 16, 11, 21, 5, 19, 2, 17, 24, 9, 22, 7, 4, 10, 8,23, 20, 3, 13, 15

Write the unscrambled message.

The 1876 election betweenRutherford Hayes andSamuel Tilden was one ofonly three times in U.S.history that a candidatewon the popular vote butlost in the ElectoralCollege. The telegram hadan offer to sell Florida’selectoral votes and therebyswing the 1876 electionfrom Hayes to Tilden.

FYI

Figure 1.2. The Pelton telegram.



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MMOW Chapter 1 - COMAP · prestigious award for a young mathematician, the Adams Prize, for the ... Maxwell and geometrician Sir William Hodge. Although cryptographers will use